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Tunable acoustic double negativity SUBJECT AREAS: Z. Liang1, M. Willatzen2,J.Li1 & J. Christensen3 CONDENSED-MATTER 1Department of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Hong Kong, 2Mads Clausen Institute, University of Southern Denmark, Alsion 2, DK-6400 Sønderborg, Denmark, 3IQFR - CSIC Serrano 119, 28006 MATERIALS SCIENCE Madrid, Spain. PHYSICS

Man-made composite materials called ‘‘’’ allow for the creation of unusual propagation Received behavior. Acoustic and elastic metamaterials in particular, can pave the way for the full control of in 16 August 2012 realizing cloaks of invisibility, perfect lenses and much more. In this work we design acousto-elastic surface modes that are similar to surface in metals and on highly conducting surfaces perforated by holes. Accepted We combine a structure hosting these modes together with a gap material supporting negative modulus and 11 October 2012 collectively producing negative dispersion. By analytical techniques and full-wave simulations we attribute the observed behavior to the and being simultaneously negative. Published 14 November 2012 lassical such as sound and light have recently been put to the test in the challenges for cloaking objects1–8 and realizing negative refraction9–18. Those concepts are just a few of recent fascinating phe- Correspondence and nomena which are consequences of artificial electromagnetic (EM) or acoustic metamaterial designs. C 19 20 Perfect imaging or enhanced transmission of waves in subwavelength apertures are other disciplines within requests for materials the scope of metamaterials which have received considerable attention both from a theoretical and experimental should be addressed to point of view21–24. Smith et al. designed an EM composite material in which the electric and magnetic response act J.C. (johan. simultaneously to exhibit a negative effective index of refraction band25. The idea was to tune the of a christensen@gmail. periodic array of split ring such that the negative permeability would occur below the fre- com) quency of a metallic rodded medium where the electric response is negative26,27. An advanced version of this design is provided by the so-called fishnet structure. In its various build-ups it contains a high figure-of-merit and gives rise to in the near-IR10, microwave28 and optical regime13. Manipulating sound waves is readily possible by utilizing metamaterials. Those materials are realized by resonating building blocks that are fabricated on a size scale smaller than the of the irradiated . It is the ability to control and tune those meta-atoms, which forms the basics in the design for tailored and unusual acoustic material responses. There are different strategies in creating negative refraction using acoustic metamaterials. One approach is to use perforations. An isotropic negative index can be obtained by coiling up small channels of perforations17. Hyperbolic metamaterial for broad-angle negative refraction on the other hand, can be constructed using layered holey structures18. Another approach is to employ locally resonating structures. The first metamaterial fabricated to possess double negativity (simultaneous negative effective bulk modulus 1/k and mass density r) was a one-dimensional tube design consisting of periodic interspaced open side branches and membranes29. Both of those resonators describe a Drude-like behavior for the effective bulk modulus and mass density respectively.

Results Designer acousto-elastic surface modes. To go beyond the lumped element design we introduce structured metallic components which are able to convey tunable double negativity behavior in a planar configuration although extensions into 3 dimensions should not add any difficulties. For the design of such a material we take a close look at the EM fishnet metamaterial and initially aim at converting its plasmonic interpretation into acousto-elasticity13. In its basic design we have two adjacent holey films made out of good conductors (Au, Ag) which are separated by a spacer. Metal surfaces punctured with subwavelength holes allow for the EM waves to penetrate much in the same way as a polariton30. Interestingly does the effective take the same plasma form whether they are supported in metals such as silver or in perforated highly conducting surfaces. It is this behavior which describes the electric response of a fishnet structure that is generated in the holey films. We will start out examining the acoustic analogy of these designer surface modes and

SCIENTIFIC REPORTS | 2 : 859 | DOI: 10.1038/srep00859 1 www.nature.com/scientificreports

Figure 1 | Transmittance and effective mass density spectra of the structured and elastically filled rigid screen. For all simulations, the normalized 3 screen thickness is fixed at h/dx 5 0.7 and the filling material has r 5 2300 kg/m and a longitudinal cl 5 1410 m/s. Furthermore we 3 consider the sample to be immersed in water, c0 5 1481 m/s and r0 5 1000 kg/m . The transmittance (b) and effective mass density Re(reff) (c) spectra are calculated for ax/dx 5 0.5. In (d,e) we hold c0/ct 5 2. The full (dotted) lines represent data obtained by modal expansions (COMSOL simulation). The incident sound plane wave is impinging at the normal direction. (f) The band diagram is simulated for a structure with the parameters ax/dx 5 0.5 and h/dx 5 0.7 and contains filling parameters as in the latter example though with c0/ct 5 3.7. later study the effective bulk modulus which is showing resemblance localized, we fill these indentations with an elastic material as ren- to the magnetic response of the fishnet structure. dered in Fig. 1a. We begin with a simple model that is constructed for Let us suppose that we have a perfect rigid screen into which holes a structure with translational invariance in the y direction and tex- are carved. In order for an incident airborne sound field to be highly tured by periodic slits of lattice constant dx along the x axis. An

SCIENTIFIC REPORTS | 2 : 859 | DOI: 10.1038/srep00859 2 www.nature.com/scientificreports incident wave excites elastic cavity modes inside the filled indenta- based on a retrieval technique31. The theoretical formalism used to tions which predominantly are governed by a displacement uz in the calculate the scattering coefficients is based on the modal expansion direction perpendicular to the lattice. The displacement generally of the and velocity fields in free-space such as the displace- satisfies the homogeneous for an isotropic : ment and stresses inside the elastic inclusion32. Even though, in the former, the metamaterial halfspace is treated in the effective medium 2 2 2 2 2 L uz L uz L uz L uz L ux l ? r ~m z zðÞlzm z , ð1Þ limit ( ax) by solely introducing specular reflections and taking Lt2 Lz2 Lx2 Lz2 LxLz into account only the fundamental elastic cavity mode, we conduct simulations by considering all higher expansion orders. As given in where l, m and r are the modulus of incompressibility (first Lame´ the caption of Fig. 1, we hold the screen thickness h/dx that is normal- coefficient), modulus of rigidity (second Lame´ coefficient) and the ized to the lattice dx fixed, but vary the width ax/dx of the indentations solid mass density respectively. As the elastic inclusions are clamped and also the inclusion material by different values of the transversal 5 5 to the rigid frame, uz(x 0, ax) 0 must vanish at the edges of the speed of sound ct. Transmittance spectra are plotted and compared to slit. Furthermore, we must guarantee zero shear- at the free- the real part of the effective mass density Re(reff) as depicted in space and metamaterial interface as waves only propagate longit- Fig. 1b and Fig. 1c respectively. We clearly see that by lowering ct udinally in a , sxz 5 0. Given the long wavelength limit which we also lower the onset of the first allowed band which means that ? we are mainly interested in, i.e., l ax, we take the fundamental the spectral locations of the acousto-elastic plasma cavity eigenmode to be dominant as it penetrates the deepest into the shift towards longer l/dx. These are the locations at elastic inclusion, hence, we can write the out-of-plane displacement 2pc0 c0 lp~ ~2ax where the effective mass density, as seen in field inside this region as: vp ct p Fig. 1c, changes sign - the transition between an opaque and trans- ~ : : iwt uzðÞx,z A sinðÞbzz sin x e : ð2Þ parent effective fluid. Fig. 1d and Fig. 1e confirm that the acousto- ax elastic plasma oscillation can be further controlled by the indentation width a . With a specific chosen filling material, in this case c /c 5 Disregarding in-plane displacement (ux 5 0) is a plausible assump- x 0 t tion in narrow elastic channels, hence we substitute Eq. (2) into Eq. 2.0, we predict that the cut-off wavelengths now locate at lp 5 4ax.In order to verify the analytical approach based on modal expansions (1), whereupon we solve for the bz comprising longit- udinal and transverse wave motion: (full lines in Fig. 1), we have further conducted full-wave simulations sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (COMSOL Multiphysics, dotted lines) and found very good agree- rffiffiffiffiffiffiffiffiffiffiffiffiffi ment. In addition we calculate the radiative transmittance band dia- r m p2 1 v v2 b ~v 1{ ~ 1{ p , ð3Þ gram which has been plotted in units of the plasma . This is z z 2 2 2 l 2m r ax v cl v rendered in Fig. 1f (see the captions for details) where the low fre- p quency plasmon-like mode interestingly shows very little dispersion which possesses a cut-off frequency vp~ct : As we are focusing on with the parallel kx of the incident sound. Again it must ax be noted that the cut-off can be controlled at will by the size of the the long wavelength regime we take into consideration only specular indentations and the choice of the filling material such as polymers. reflection when imposing continuity of an normally incident plane This allows for the creation of designer surface states at almost arbit- wave irradiating onto the structured half-space (h R ‘). From here it rary frequency. is straightforward to write down expressions for the reflection co- efficient and the effective impedance: Tunable negative effective bulk modulus. The magnetic response of the aforesaid EM fishnet structure is induced by the component Zeff {1 b ðÞlz2m R ~ , Z ~r c ~ z , ð4Þ consisting of metal-insulator-metal (MIM) units. These structures 0 Z z1 eff eff eff S v eff f are hosting which can give rise to negative values of the 8 a effective permeability. Recently a theoretical study on the acoustical where the structure factor S ~ x : Substituting Eq. (3) into Eq. (4) f p2 d version of the MIM structure showed that these sandwich-materials x would give rise to an effective negative bulk modulus 1/k, which su- and rewriting the effective speed of sound as ceff 5 k0/bz,wearriveat: ! stains for a broad range of angles of incident sound33. The conse- 2 2 quence of a negative bulk modulus in a fluid element is the overall p dxrc0 vp r ~ 1{ , ð5Þ expansion of it as a reaction to a positive acoustic stimulus. This eff 8a v2 x behavior has recently been observed experimentally in the audio-fre- quency range34. In Fig. 2a we show the slitted version of this design when solving for r , which is the canonical Drude form of an eff and compute the normally incident sound transmittance through it acousto-elastic surface mode with a geometrical ‘‘plasma’’ frequency by the same method as reported in33. We fill a fluid in the interme- of diate gap region of the structure (with height hg) and study the con- p trollability of the resonance by changing the value of the speed of vp~ct , ð6Þ ax sound cg. We compare the spectral transmittance (Fig. 2b) to the retrieved effective bulk modulus Re(1/k) (Fig. 2c) that at resonance the cut-off frequency of the elastic waveguide. In summary we can say sustain negative values over extended ranges. These regions of nega- that structured rigid screens will have this form of the effective mass tive 1/k are as expected accompanied by polaritoniclike bandgaps, density provided all dimensions are treated in the effective medium which in Fig. 2b are seen by the presence of sharp dips in the limit. The acousto-elastic cut-off frequency can be modified at will transmittance spectra. Moreover, we predict a distinct correlation such that the screen can constitute a tunable high-pass filter with a between these gap induced resonances and the speed of sound cg stopband from zero frequency up the cut-off. therein as the resonating modes shift towards lower frequencies In the following we will take a look at the tunability of the struc- when the filling material speed-of-sound is lowered. tured screen with the aim to design the spectral location of the acoustic ‘‘plasma’’ frequency. In order to do so, we employ a rigid Composite material exhibiting double negativity. The aforemen- structured screen of finite thickness h as depicted in Fig. 1a and tioned tuning strategy will prove to be very useful at the stage when calculate complex transmission and reflection coefficients for using merging the previous Drude-like metamaterial containing negative them to numerically obtain the effective constitutive parameters reff together with the present one to form a composite demonstrating

SCIENTIFIC REPORTS | 2 : 859 | DOI: 10.1038/srep00859 3 www.nature.com/scientificreports

double negativity. The pursuit is to shift these bulk modulus 1/k resonances shown in Fig. 2, below the cut-off frequency vp of the elastomer filled screen. This particular design idea is a utile extension of the former one, we solely need to fill the slits with an elastic material as sketched in Fig. 3a. We design a plasma frequency to be located at lp < 2.8dx below which the effective mass density is negative. We have chosen to use a fluid inside the gap of speed cg/c0 5 0.25 such that the negative response of 1/k occurs within the forbidden region, below the plasma frequency. The combination of simultaneous negative constitutive parameters induces a passband of full sound transmission at around l < 4.4dx, as seen in Fig. 3b. Here again we compare analytical predictions with full-wave simulations where a fairly good agreement is obtained, confirming propagation at a region of double negativity. Interestingly we observe negative dispersion in this propagation band which has been computed by the transmittance as a function of frequency and parallel momen- tum, as rendered in Fig. 3c.

Discussion We have demonstrated both by analytical and numerical simulations that structured rigid screens filled with elastic inclusions can be described in the long wavelength limit as an effective medium char- acterized by a mass density of the plasmon form. This type of meta- material is thus not only used to control surface waves but also to create them. The combination of this behavior together with gap materials hosting bulk modulus resonances, produces negative disper- sion with simultaneous negative values of reff and 1/keff,whichisa Figure 2 | Transmittance and effective bulk modulus Re(1/k) spectra for characteristic of negative materials. The near-field two facing structured rigid plates with a/dx 5 0.5, h/dx 5 0.2 and the gap coupling of the constituting resonances can be considered weak, which separation hg/dx 5 0.1. The gap is filled with a fluid of relative mass density is essential when tuning the two effective parameters independently. rg/r0 5 1 and speeds of sound cg/c0 as indicated. The transmittance is presented in logarithmic scale and calculated at normal incidence.

Figure 3 | Acoustic double negativity band in a composite metamaterial. (a) The composite metamaterial is designed with parameters a/dx 5 0.5, h/dx 5 3 0.5 and hg/dx 5 0.1 with an elastic inclusion containing r 5 2300 kg/m , cl 5 1400 m/s and ct 5 530 m/s. (b) With cg/c0 5 0.25 the transmittance spectra (right axis) is accompanied with plots of Re(1/keff) and Re(reff) (left axis). The full (dotted) lines represents data obtained by modal expansions (COMSOL simulation). (c) Band diagram simulation for the structure with parameters as given in (b).

SCIENTIFIC REPORTS | 2 : 859 | DOI: 10.1038/srep00859 4 www.nature.com/scientificreports

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B 75, 195447–195451 (2007). double negativity metamaterial. Sci. Rep. 2, 859; DOI:10.1038/srep00859 (2012).

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